D.J. Hill, C. Pantano and D.I. Pullin- Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock

8/3/2019 D.J. Hill, C. Pantano and D.I. Pullin- Large-eddy simulation and multiscale modelling of a RichtmyerMeshkov instability with reshock

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J. Fluid Mech. (2006), vol. 557, pp. 2961. c 2006 Cambridge University Pressdoi:10.1017/S0022112006009475 Printed in the United Kingdom

29

Large-eddy simulation and multiscale modellingof a RichtmyerMeshkov instability with reshock

B y D . J . H I L L , C . P A N T A N O A N D D . I . P U L L I N

Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, [email protected]

(Received 6 July 2005 and in revised form 4 November 2005)

Large-eddy simulations of the RichtmyerMeshkov instability with reshock are pre-sented and the results are compared with experiments. Several configurations ofshocks initially travelling from light (air) to heavy (sulfur hexafluoride, SF6) have beensimulated to match previous experiments and good agreement is found in the growthrates of the turbulent mixing zone (TMZ). The stretched-vortex subgrid model used inthis study allows for subgrid continuation modelling, where statistics of the unresolvedscales of the flow are estimated. In particular, this multiscale modelling allows theanisotropy of the flow to be extended to the dissipation scale, , and estimates to beformed for the subgrid probability density function of the mixture fraction of air/SF6based on the subgrid variance, including the effect of Schmidt number.

1. Introduction

During the process of refraction as a shock interacts with an interface separatingtwo gases, vorticity is, in general, deposited at the interface by means of baroclinictorque. The physical mechanism of this vorticity production is the miss-alignment ofthe pressure gradient across the shock and the local density gradient at the contactduring shock passage. The rotational flow associated with this localized vorticitycauses the interface to distort strongly in areas of maximal misalignment, formingcharacteristic structures such as bubbles and spikes. The growth in the interfaceamplitude resulting from the shockinterface interaction is generally referred to asthe RichtmyerMeshkov instability (RMI) (Richtmyer 1960; Meshkov 1969) and issometimes considered to be the impulsive limit of a RayleighTaylor instability. Theanalysis of Richtmyer (1960) showed that initially small perturbation amplitudes

grow linearly with time, and other work such as Mikaelian (1989) suggests that evenwhen the perturbation is nonlinearly saturated, the growth continues to be linear intime. The structures generated by the RichtmyerMeshkov instability are themselvessubject to vortex pairing and additional instabilities such as the KelvinHelmholtzinstability, leading to a wide range of physical scales in the area of the interface.

The RichtmyerMeshkov instability plays a fundamental role in the context ofmany physical settings, both natural and man-made. Evidence of the RMI has beenseen by the Hubble Space Telescope in remnants of the explosion of supernova1987A (Sonneborn et al. 1999; Maran et al. 2000). In related stellar events, the RMIis used to explain the overturn of the outer portion of collapsing cores of supernovas

(Smarr et al. 1981). Proposals have been made to exploit the mixing properties ofthe instability in supersonic combustion engines (Yang, Kubota & Zukoski 1993).Conversely, in the context of inertial confinement fusion, the RMI-induced mixingbetween fuel and capsule is a liability which provides considerable challenges in


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30 D. J. Hill, C. Pantano and D. I. Pullin

achieving the required compression (Lindl, McCrory & Campbell 1992). This lastexample alone has motivated considerable interest in the fundamental aspects of theRMI; recent reviews of the RMI include Zabusky (1999) and Brouillette (2002).

This paper examines, by computational simulation, a canonical RichtmyerMeshkov instability realized within the confines of a shock-tube as a planar shock

interacts with a co-planar density interface formed by the contact between two gases,air and sulfur hexafluoride (SH6) (Vetter & Sturtevant 1995). Slight perturbations orirregularities in the density interface, for example local deviations from co-planarity,form the density misalignments required to initiate RMI during shock interaction. Asecond and much more energetic RMI occurs after the initial shock has traversedthe extent of the shock-tube, reflected off the tube end and reshocks the now greatlydistorted interface.

The double-shock process produces a large dynamical range of turbulent scales, re-quiring, with presently available computational resources, the techniques of large-eddysimulation (LES). This approach entails a loss of detailed information contained in

fine unresolved scales whose dynamical interaction with the computationally resolvedscales is only modelled. Because mixing of the two gases in the turbulent zone is anessentially small-scale (subgrid) process, conventional LES, with its exclusive focus onresolved-scale transport, cannot accurately capture this important aspect of the flow.Owing to its structural ansatz, using vortices that are local (asymptotic) solutions tothe NavierStokes equations at the subgrid level, the stretched-vortex model enablesestimation of the contribution to certain statistical quantities from scales below theresolved-scale cutoff. This, together with resolved-scale information, allows somedegree of truly multiscale modelling of the flow, including predictions of mixing.

Apart from the physical modelling issues associated with LES, simulation of com-pressible turbulent flows demands two mutually exclusive numerical approaches. Onone hand, the presence of shocks in three-dimensional flows, whose length scale is ofthe order of the mean free path, implies that the numerical method must be of a shock-capturing type. On the other hand, turbulence is better simulated when the numericalmethod is non-dissipative. Since all shock-capturing methods are dissipative, two mu-tually orthogonal numerical requirements arise. To address this difficulty, we develo-ped hybrid numerical methods that are shock-capturing around discontinuities(shocks) and revert to centred non-dissipative discretizations in the remaining regionsof the flow, including those encompassing the regions of turbulent activity. The presentmethod is an extension of the hybrid centred-upwinded algorithm of Hill & Pullin(2004).

The target experiment of Vetter & Sturtevant (1995) and the modelling of thelaboratory initial and boundary conditions used at present are discussed in 2. Therefollows, in 3, a short description of the relevant filtered NavierStokes equationsand the stretched-vortex subgrid-scale (SGS) model used in the simulations. In 4, wegive an account of the hybrid numerical scheme employed. The main resolved-scalesimulation results are presented in 5. The multiscale modelling based on subgridcontinuation is described in 6 and 7. This includes predictions of subgrid mixingproperties and estimates of the effect of Schmidt number on both scalar spectra andscalar probability density functions (p.d.f.).

2. Flow description

We discuss, in the following order: the experimental flow conditions, simulationboundary conditions, the treatment of the sidewalls of the shock-tube, the endwall of


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Large-eddy simulation of a RichtmyerMeshkov instability 31

Unshocked Unshocked

SF6airair

Shocked

Periodic boundariesReflecting wall

x

y

z

Shock

0.20 m 0.62 m

0

.27m

Figure 1. Geometry of the simulation domain for the Mach 1.50 case, indicating the locationof the mathematical origin relative to the reflecting endwall. Note the full simulation domain issquare in (y, z)-cross-section with periodic boundary conditions also applied at y, z =

0.135 m.

Window

0.60.50.40.30.20.1

t(ms)

0.1 0.2 0

1

0

2

3

4

5

6

Air SF6

shock

x (m)

interface

expansion

Figure 2. Approximate wave diagram for the interaction of a M= 1.50 shock with the air/SF6

interface. Both the exit of the initial reflected shock and the primary shock are indicated, as isthe field of view for the flow-visualization system.

the shock tube and the open end of the tube and, finally, the initial configuration. Aschematic of the initial flow configuration, including the incident shock, the perturbedinterface between the two test gases, and other features of the geometry and boundaryconditions, is shown in figure 1. Figure 2 shows a one-dimensional wave-diagram ofthe RMI in an (x, t)-plane, including the first interaction of the shock with theinterface, shock reflection off the tube endwall, reshock, and the interaction of areflected expansion (resulting from reshock) with the gas interface.

2.1. Experimental domain and parameters

For the purpose of validation, we have chosen for numerical investigation the air/sulfur hexafluoride (SF6) RichtmyerMeshkov variable-length shock-tube experiments


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IIb VIb VIIb

Incident Mach number 1.24 1.50 1.98Pressure (kPa) 40 23 8Length, membrane to endwall (cm) 110 62 49Instantaneous velocity (m s1) 72 150 287Shocked growth rate (m s1) 2.1 4.2 7.5Reshocked growth rate (m s1) 17.0 37.2 74.4Shocked observation times (ms) 4.76.7 2.23.2 1.72.5Reshocked observation times (ms) 15.516.5 4.05.5 1.72.5

Table 1. The test conditions and growth rates of the interface thickness from Vetter &Sturtevant (1995). These experiments correspond to IIb, VIb, and VIIb therein and theobservation times are approximate as they result from figure 6 of that work.

Property Air SF6

Molecular mass (kg kmol1

) 29.04 146.07Ratio of specific heats: 1.40 1.09Density ( kg m3) 1.18 5.97Kinematic viscosity (106 m2 s1) 15.7 2.47Prandtl number 0.71 0.90Diffusion coefficient in air (106 m2 s1) 20.4 9.7

Table 2. Gas properties of air and SF6 at 25C and 1 atm, as used in the simulation.

of Vetter & Sturtevant (1995). In these experiments, a shock in air travelled down thetube with a Mach number strength ranging from 1.18 to 1.98. The shock encountereda contact surface in the form of a thin membrane interface separating the air from theremainder of the tube filled with SF6. The passage of the transmitted shock induceda mixing zone between the air and SF6 which was instantaneously accelerated toreported velocities of 56 m s1 to 287ms1, depending on the case. Using either spark-schlieren photography or high-speed motion pictures, data were recorded as the mixingzone passed the field of view of an observation port, and data were again recorded afterthe transmitted shock reflected off the closed end of the shock tube and reshockedthe mixing zone, bringing the mixing zone back into the observation field and reducingits mean velocity substantially. The length of the shock tube from the interface to theendwall was adjusted prior to each experiment to ensure that the reshocked mixing

zone would develop in view. Vetter & Sturtevant (1995) measured the instantaneouswidth of the mixing zone and calculated two mixing-zone growth rates, one after theinitial shock and a second following reshock, for each experiment. These growth ratesserve as the primary comparison between the experiments and the present simulations.

Owing to diagnostic limitations, the experiments with single-spark schlieren allowedfor very few measurements of the mixing-layer width as the mixing zone passed theobservation port. For this reason, the comparisons in this paper focus on three of theexperiments in which high-speed motion picture photography was used, as the growthrates are ostensibly more accurate. These experiments correspond to Mach numbers1.24, 1.50 and 1.98 and are referred to as cases IIb, VIb and VIIb, respectively. Table 1

summarizes the experimental configurations and results. In the LES, we match thephysical dimensions of the experimental domain and the particular gas properties ofair and SF6, as summarized in table 2. The shock tube itself was square in cross-section with dimensions 0.27 m 0.27 m and of variable down-tube length. More


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precisely, in right-handed Cartesian coordinates, x = (x1, x2, x3) = (x , y , z), the positivex-axis aligns with the down-tube direction with x = 0 at the initial gas interfacetowards the reflecting wall. The simulation domain was 0.20 m6 x6L0.135m6y6 0.135m0.135m6 z6 0.135 m where L is 1.10 m, 0.62 m or 0.49 m for differentruns. A small portion of the tube with x < 0 is included in the computation, so that

reflected and transmitted shocks will be essentially planar when they exit the domain.2.2. Modelling boundary conditions

To facilitate interpretation of statistical results, the sidewalls of the shock tube inthe simulations were replaced with periodic boundary conditions, thus allowing thecalculation of planar spectra in quantities of interest and providing two statisticallyhomogeneous directions. In making such a boundary approximation, the non-uniformboundary layer flow at the physical walls is absent from the simulation, but this is notfound to be critical in comparing with the experiment. In fact, because such boundaryeffects lead to ambiguities in the interpretation of flow-visualization photographs,Vetter & Sturtevant (1995) explicitly designed their experiment to ensure that the

shock-wave/boundary-layer interaction would not play a dominant role in the mea-sured growth rates.

The endwall of the shock tube (located at x = L) is, computationally, a reflectingboundary. No wall model or other special treatment was required in the large-eddysimulations at the reflecting end of the tube. The measured widths and approximatelocations of the mixing zone suggested that the turbulent region does not reach thereflecting wall, an assumption which was born out in our simulations. In the computa-tional domain, the shock tube is truncated with an open end. The treatment of thisopen end models the effects of the unsimulated remainder of the physical tube. Toallow for stable long-time integration after the reflected shock has exited the computa-

tional domain, the density, velocity, and pressure are prescribed in the form of charac-teristic boundary conditions (Thompson 1987) at the exit plane.

2.3. Modelling initial conditions

The flow is initialized in three regions distributed from left to right: a post-shockstate in the air side consistent with a shock travelling at the corresponding Machnumber in the positive direction towards the contact between the two gases, a regionof unshocked stationary air, and finally unshocked stationary SF6. The quiescentgases are assumed to be in thermal equilibrium and under uniform pressure. Inagreement with the experiments, this initial pressure is taken, for each case, to be40 kPa, 23 kPa, or 8 kPa (see table 1) and a room temperature of 286K was assumed.Taken together, these choices determine the complete state of the unshocked gases. Aschematic of the initial flow configuration, including the incident shock, the perturbedinterface between the two test gases, and other features of the geometry and boundaryconditions, is shown in figure 1.

The shock is initialized at a location of0.05m in the x-direction and the lateralcentroid of the perturbed contact at x = 0.0 m leading to an initial shock interactionwhich instantaneously accelerates the contact. Vetter & Sturtevant (1995) indicate thata relatively weak expansion wave from the driver section of the shock tube follows,0.5 ms in the Mach 1.5 case, after the shock. In single-mode Mach 1.5 experiments withthe same shock tube and gases Prasad et al. (2000) also observe this expansion and

indicate that its amplitude was sufficiently small as to be ignored in the presentationof their results. In light of both this and the fact that insufficient details were presentedfor the cases of interest, this expansion wave was neglected in the initial conditions forour simulations. Also, no attempt was made to model the effects of the composition


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of the thin (0.5m) nitrocellulose membrane, but the shape was represented as aperturbed interface. These perturbations result from the wire mesh that was installedto support the membrane and they provide spatial variations in density which in turngive rise to the RichtmyerMeshkov instability. The wire mesh formed a regular gridwith spacing of 1 cm in both the horizontal and vertical directions (the wire diameter

was 0.23 mm). Prior to each experiment, the membrane was pushed into the meshwith an estimated amplitude of about 1 mm. Vetter & Sturtevant (1995) showed thatby varying the order of the vertical and horizontal meshes and the membrane, thepre-reshock growth rates could vary by an order of magnitude. It is unclear to whatextent this is due to the additional mixing of the membrane or to the sensitivity to theinitial interface shape. Additionally, Greenough & Burke (2004) have demonstratedthat the initial density perturbation spectrum can have a strong effect even on thenonlinear stages of mixing-layer growth. In light of the uncertainty in the initialconditions and the role of the membrane, the actual width of the mixing zone is notexpected to compare well during the observation times prior to reshock.

An interface representation similar to that employed by Cohen et al. (2002) is usedhere to incorporate small modes, ostensibly produced by the wire mesh, and additionalsymmetry-breaking modes which model the more random smaller-scale irregularities.The simulation interface was taken to be the linear combination of a regular egg-carton perturbation with a much smaller irregular perturbation. Mathematically, theinterface was prescribed as follows:

xI(y, z) = a0| sin(y/) sin(z/)| + a1h(y, z), (2.1)where the first term, the so-called egg-carton, | sin(y/) sin(z/)|, models theregularity of the mesh and the second, h(y, z), takes the form of a symmetry breaking

perturbation with random phase, but with a prescribed initial power spectra of theform k4exp((k/ ko)2). The function h(y, z) was computed once and saved so that itcould be used for all the runs with the same computational resolution.

The fundamental carton wavelength, , was taken to be 27 m/14 0.02 m as a com-promise between the actual grid spacing of 0.01 m and the desire to allow enough res-olution so that each perturbation could evolve into the nonlinear regime. Cohen et al.(2002) found it necessary to strike a similar balance. In the non-symmetric portionh(y, z), the parameter ko = 4 was chosen, which corresponds to a peak wavelengthof (/

8) m, and values of the coefficients a0 and a1 were 0.25cm and 0.025 cm,

respectively.

3. Equations of motion and subgrid modelling

3.1. Two-component Favre-filtered NavierStokes equations

The Favre-filtered (i.e. density weighted) NavierStokes equations provide a naturalseparation of the large scales to be simulated from the small scales to be modelled(Zang, Dahlburg & Dahlburg 1992). We denote Favre-filtered quantities by

f =f

, (3.1)

where f is an arbitrary field, is the density and the overbar indicates the filteringoperation

f(x) =

G(x x )f(x) dx, (3.2)


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with convolution kernel G. The filtering procedure combined with some modellingassumptions (e.g. negligible subgrid viscous work and triple correlations) leads to thefollowing LES equations of motion for the density , momentum ui , total energy E,and a scalar representing mixture fraction . The dimensional conservation transportequations are

t+

uj

xj= 0, (3.3a)

ui

t+

(ui uj + pij )

xj=

dij

xj ij

xj, (3.3b)

E

t+

(E + p)ujxj

=

xj

T

xj

+dj i ui

xj q

Tj

xj, (3.3c)

t+

(uj )

xj=

xj D xj q

j

xj, (3.3d)

where repeated indices denote summation and the subgrid terms

ij = (ui uj ui uj ), (3.4a)qTj = (

cpT uj cpTuj ), (3.4b)q

j = (uj uj ), (3.4c)

represent subgrid stress tensor, and the heat and scalar transport flux, respectively.The filtered total energy E contains the subgrid kinetic energy and is given by

E =p

( 1) +12

(uk uk) +12

kk , (3.5)

while the filtered pressure, p, is determined from the ideal equation of state for amixture of gases,

p =RT

m, (3.6)

where R is the ideal gas constant; we have neglected temperaturespecies compositioncorrelations in (3.6). This equation, together with (3.5), defines the Favre-filtered

temperature T as a function of E, kinetic energy and the composition of the mixture.As there are only two gases, the single scalar is sufficient to specify the local mixturecomposition. Formally, air is assumed to behave as a single species with the averagemolecular weight of air. Within these assumptions, takes the value 0 in the air sideand 1 in the SF6 side. The mass fraction of air is a = 1 , likewise for SF6 s = .The mean molecular weight is then given by 1/m = (a/ma + s /ms ), where ma andms denote the molecular weights of air and SF6, respectively.

Transport properties of the mixture are determined from binary mixing rules andthe pure component transport properties (Reid, Pransuitz & Poling 1987). In the caseof viscosity (similarly for heat conduction, , and diffusivity, D), each gas obeys

l = ol (T /To)

0.786

, with l = a, s (air and SF6, respectively), then

=a am

1/2a + s s m

1/2s

am1/2a + s m

1/2s

. (3.7)


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The deviatoric Newtonian stress tensor dij of the mixture with appropriately filteredquantities is then expressed as

dij =

ui

xj+

uj

xi

2

3

uk

xkij

. (3.8)

Finally, the average specific heat ratio is defined by

=cp

cv, (3.9)

where the specific heat capacity at constant pressure is given by

cp = cp,a a + cp,s s , (3.10)

and cv = cp R/m. The element heat capacities, cp,l , are assumed to be independentof temperature in the present study.

The filtering procedure described in (3.2) is purely formal. It is not, and cannotbe, performed in LES, unless we have the full DNS (or experimental) fields in hand,

in which case LES is irrelevant. Here and hereinafter, we identify variables definedformally by (3.1) and (3.2) with resolved-scale quantities in actual LES. This is strictlya resolved-scale modelling assumption at the level of those for subgrid quantitiesdescribed below. For a discussion of this and other conceptual foundations of LES,see Pope (2004).

3.2. Application of the stretched-vortex subgrid model

Closure of the LES equations is completed in the form of a model for the subgrid inter-action terms: stress tensor, ij , turbulent temperature flux, q

Ti , and mixture fraction

flux, qi . This is achieved by using the stretched-vortex model, originally developed

for incompressible LES by Misra & Pullin (1997), but extended to compressible flows(Kosovic, Pullin & Samtaney 2002) and subgrid scalar transport (Pullin 2000). Inthis model, the flow within a computational grid cell is assumed to result from anensemble of straight, nearly axisymmetric vortices aligned with the local resolvedscale strain or vorticity. The resulting subgrid stresses are

ij = k

ij evi evj

, (3.11a)

qTi = c2 k1/2

ij evi evj(cpT)

xj, (3.11b)

qi =

c

2

k1/2ij evi evj

xj, (3.11c)

where k =

kcE(k) dk is the subgrid energy, ev is the unit vector aligned with the

subgrid vortex axis, = / is the kinematic viscosity and kc =/c represents thelargest resolved wavenumber. This subgrid turbulent kinetic energy, k, is estimatedby assuming a spiral vortex of the Lundgren (1982) form, whose energy (velocity)spectrum for the subgrid motion is given by

E(k) = K02/3k5/3exp[2k2/(3|a|)], (3.12)

where K0 is the Kolmogorov prefactor, is the local cell-averaged dissipation (resolvedflow plus subgrid scale) and a = Sij e

vi e

vj is the axial strain along the subgrid vortex

axis whereSij =

1

2

uixj

+uj

xi

, (3.13)

denotes the resolved rate-of-strain tensor.


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To complete the model, the group prefactor K02/3 must be calculated for each cell

from the resolved flow. This is done by a structure function matching (Lesieur &Metais 1996; Voelkl, Pullin & Chan 2000; Pullin 2000). Essentially, the second-ordervelocity structure function F2(r) when averaged over the surface of a sphere of radius gives

F2() = 4

0

E(s/)

1 sin ss

ds. (3.14)

The spectra, (3.12), and the assumption that the exponential can be ignored whenevaluating at the separation scale give the group prefactor as

K02/3 =

F2()2/3A

, (3.15)

where A = 4

0s5/3(1 sin s/s) ds 1.90695. In practice, and c are taken to be

the grid spacing, x, and the spherical average of the structure function is computedas a local estimate using a six-point stencil on the resolved scales

F2() = 16

3j =1

u+1 2 + u+2 2 + u+3 2 + u1 2 + u2 2 + u3 2j , (3.16)

where ui = ui (xo ej ) ui (xo) denotes the ith velocity component difference inthe unitary direction ej at the point xo.

This subgrid model is based on subgrid elements in the form of spiral vortices thatare local approximate solutions of the NavierStokes equations (Lundgren 1982), andthe scalar transport equations (Pullin & Lundgren 2001) for a constant-density fluid.It has been applied to two-fluid mixing in RayleighTaylor instability (Mattner, Pullin& Dimotakis 2004) where the process of subgrid mixing of a variable-density fluidis modelled, in the sense of (3.11), via an SGS temperature flux treated as a passivescalar. There is, therefore, no explicit model representation of subgrid baroclinicvorticity production. Insofar as the SGS vorticity spectrum is eschewed in favour ofthe SGS energy spectrum, this deficiency may not be fatal. The effect of small-scaletwo-fluid mixing on the velocity spectrum at large wavenumbers and large Reynoldsnumbers, remains an open question.

4. Computational approach

4.1. Numerical methodAn improved version of the TCD-WENO hybrid method (Hill & Pullin 2004), basedon tuned centre-difference (TCD) stencils is used to integrate the equations of motionin conjunction with a third-order strong-stability preserving (SSP) RungeKutta time-stepping scheme (Gottlieb, Shu & Tadmor 2001). The spatial discretization has beenconstructed explicitly to be shock-capturing, to enforce weak convergence (predictionof the correct shock speeds), and to revert smoothly to a centred stencil with lownumerical dissipation and good wave-dispersion properties in turbulent-flow regionsaway from shocks. Here, the hybrid method uses a bandwidth optimized five-pointcentre-difference stencil tuned for better modified wavenumber behaviour at the price

of a reduction in the order of accuracy from fourth-order to second-order. Thistuning was achieved by minimizing the spatial truncation error (Ghosal 1996, 1999)for the NavierStokes equations under model assumptions of a von Karman spectra.In Hill & Pullin (2004), the TCD scheme was shown to work well on 323 LES of


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decaying compressible turbulence with a turbulent Mach number of Mt = 0.488 andan initial Taylor Reynolds number Re = 175. Only in thin regions containing shocksdoes the hybrid method switch to a finite-difference WENO scheme (Jiang & Shu1996) whose optimal candidate stencil has been modified to match that of the TCD.The implementation of the hybrid flux-based TCD portion of the scheme used in this

paper is the extended version presented in Pantano et al. (2005) which incorporates aflux-based representation for adaptive mesh refinement.

The present flux-based numerical method ensured discrete mass, momentum andenergy conservation with unique interpolated flux values at the grid cell wall, evenwhen the scheme switches method. To illustrate the technique, consider a uniformone-dimensional discretization of a function f(x). The fluxes, Fi+1/2 at the right-handsidewall of the ith computational grid cell, and the numerical approximation of thederivative of f(x) are related through

f

x i Dx f =Fi+1/2 Fi1/2

x

, (4.1)

where x denotes the grid spacing and Dx , the finite-difference stencil operator. Thesmallest five-point centred stencil operator has the form

Dx f = 1x

((fi+2 fi2) + (fi+1 fi1)), (4.2)where = 1/2 2 is required for at least second-order accuracy; =1/(12) is thestandard fourth-order stencil, and = 0.197 corresponds to the TCD stencil. Thedivergence-like TCD flux is given by

Fi+1/2 = (fi+2 + fi+1) + ( + )(fi+1 + fi ). (4.3)This can be verified by introducing (4.3) into (4.1). To ensure numerical stability,the discrete version of the convective term of the momentum, scalar transport andenergy equations are written in the compressible extension of the skew-symmetricform (Blaisdell 1991), such that the summation by parts property is satisfied. Thisimplies that derivatives of quadratic products of two functions f and g are actuallyevaluated as

(f g)

x 1

2(Dx (fg) + fDx g + gDx f). (4.4)

Moreover, it is possible to show that this skew-symmetric form can also be written influx form (Ducros et al. 2000). For the case of the TCD scheme, the skew-symmetricflux is given by

FT CDi+1/2 =12{[(gi+1 + gi1)(fi+1 + fi1) + (gi+2 + gi )(fi+2 + fi )] + [(fi+1 + fi )(gi+1 + gi )]}.

(4.5)

This form is used for the convective terms in the momentum, scalar and energy equa-tion (cast in the internal energy variance conserving form suggested by Honein &Moin 2004), that reads

(ui uj )

xj 1

2

(ui uj )

xj +

uj

2

(ui )

xj +

ui

2

(uj )

xj , (4.6a)

(uj )

xj 1

2

(uj )

xj+

uj

2

()

xj+

2

(uj )

xj, (4.6b)


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IIb VIb VIe VIIb

Incident Mach number 1.24 1.50 1.50 1.98Computational grid 616 1282 388 1282 776 2562 327 1282Computational resolution: x (cm) 0.21 0.21 0.105 0.21Simulation time (ms) 16.62 6.35 12.0 2.57

CPU hours 3982 972 38 400 544

Table 3. The computational cost in CPU hours for each of the runs. Simulation VI e is ahigher resolution version of VIb computed to about twice the experimental time.

((E + p)uj )

xj 1

2

(euj )

xj+

uj

2

e

xj+

e

2

(uj )

xj,

+ui

2

(ui uj )

xj+

ui uj

2

ui

xj+ p

uj

xj+ uj

p

xj, (4.6c)

where e = E/ uk uk /2 is the internal energy.Around shocks, the WENO scheme naturally computes cell wall fluxes, FWENOi+1/2 ,

based on a convex weighting of candidate stencils in an attempt to minimizeinterpolation across discontinuities. The present hybrid scheme simply selects theWENO fluxes for cells in a tight area around shocks, but uses the centred stencilselsewhere according to the following pressure and density relative curvature criteria

C = {x R3 : |p| > cx2, | | > cx2, p > 0}, (4.7)where

p = pi+1 2pi + pi1

pi+1 + 2pi + pi1, (4.8a)

=i+1 2i + i1i+1 + 2i + i1

. (4.8b)

Moreover, all grid cells in a neighbourhood of radius nx of the cells that satisfythe test condition are marked as containing the discontinuity. The three-dimensionalversion of this test is used in the simulations. The values of c and n that proved togive the best results were 2.5 103 and 5, respectively. Then, the hybrid flux takes theform

Fi+1/2 =FWENOi+1/2 in C,

FTCDi+1/2 in C,(4.9)

where C denotes the complement ofC.4.2. Simulations

Four large-eddy simulations were performed and the physical parameters chosen tomatch the experiments as summarized in table 1. Three of these were at a resolutiondefined by the grid spacing of x = 0.21 cm. They ran to the end of the experimentaltime with modest computational cost (table 3). These simulations were of sufficient

resolution to form the basis of our comparisons of mixing-layer width with theexperiments. The fourth simulation was of the Mach 1.50 case (VIe), it used twicethe resolution (i.e. x = 0.105 cm) and was integrated for twice the experimental timeto allow for a much more detailed analysis of the flow evolution and the subsequent


12/33


(a) (b)

Figure 3. Images of the mixing zone for case VIe at (a) t= 4.8 ms, (b) t=10ms.

turbulent decay of the mixing region. This case was chosen in part because of itsmodest domain size (L = 0.62 m) and the fact that it was the most throughly studiedexperiment in Vetter & Sturtevant (1995). Figure 3 shows instantaneous renderingof isosurfaces of a mixture fraction at several times during the evolution of case

Mach 1.50, exemplifying the different states of the flow.The simulations were all performed by a parallel FORTRAN90 program on QSC,an unclassified Tru64 cluster at the Los Alamos National Laboratory; each processoris an Alpha EV6 with 4GB of memory and a clock speed of 1.25 GHz.

5. Simulation results (traditional statistics)

Traditional statistical results (i.e. those of the resolved fields) obtained from the LESare summarized here. First, the mixing-layer width is defined and comparisons withthe experiments are made based on the modest resolution runs. The higher-resolutionMach 1.50 mixing-layer width is also obtained up to times that are much longer thanthe experimental time of 6.35 ms and the turbulent kinetic energy is also computed;this analysis helps to gain a better understanding of the importance of the shocks andsubsequent expansion waves that evolve in the domain. Plane-averaged flow quantitieson both the resolved scale and subgrid are presented, illustrating the low turbulentMach number in the flow and the role of the subgrid in dissipation and kinetic energy.Following this, the mixing zone is examined in the late time stages after the shockand expansion interactions. During this time, the mixing zone grows slowly at bestand exhibits some of the characteristics of decaying weakly compressible turbulence.Two statistical measurements are introduced: the instantaneous plane average, that

for an arbitrary field f is defined as

f(x, t) = 1A

f(x , y , z , t ) dy dz, (5.1)


13/33


IIb VIb VIIb

Incident Mach number 1.24 1.50 1.98Experimental final layer thickness (cm) 13.7 10.2 8.25Computed final thickness (cm) 12.2 10.6 8.0

Measurement time (ms) 16.50 6.25 2.50Simulation time (ms) 16.62 6.35 2.57First shock time in simulation (ms) 0.119 0.098 0.073

Table 4. A comparison of the mixing width at the end of the experiment with the computedmixing width at a similar time in the simulation. The experimental times and widths areapproximate as they were measured from figure 6 of Vetter & Sturtevant (1995).

where A is the y/z cross-sectional area of the tube (A= 0.272 m2), and the volumeaverage

|f(t)| = 1V f(x , y , z , t ) dx dy dz, (5.2)where V is the volume of the computational domain. Note that (5.2) is equivalent tointegrating (5.1) in the x-direction and dividing by the length of the domain.

Spectra are computed in the centre of the mixing zone to confirm that universalk5/3 scaling is recovered. Reynolds numbers and dissipation lengths are calculatedfrom the flow and the bubble and spike mixing process is analysed by examining theresolved variances. Additionally the mixing proprieties across the width of the layerare investigated.

5.1. Mixing-layer growth and comparison with experiment

To make contact with the mixing-zone growth rates estimated by Vetter & Sturtevant(1995) we define a mixing width MZ from the planar averaged mixture fraction,according to

MZ(t) = 4

tube

(1 )dx. (5.3)An analytic example can be helpful in understanding this measure better. Assumea simple average profile that represents the transition from air ( = 0) toSF6 ( = 1) and centred at xc in an infinitely long tube to be of the form= (1 + tanh(2(x xc)/ h))/2. Introducing this function into (5.3) gives the expectedmixing width MZ = h. In the actual experiments, measurements were taken from

photographs of the mixing zone, but the way in which these measurements of thewidth were made was not reported. This uncertainty tempers the degree to which wedraw conclusions in our comparison, and, more broadly, affects the comparison ofmost turbulent RMI experiments to theoretical results and computational simulations.

The computed widths of the mixing zone for the different modest-resolutionsimulations are shown in figure 4, where the experimental growth rates are plottedas solid lines of duration corresponding to the experimental observation times. Theagreement in growth rate for all three simulations are quite reasonable. As the simula-tion initial perturbations were much larger than those inferred for the experiment, noactual agreement in the mixing-zone width was anticipated in the early stages of the

simulation. The measured width during the times prior to reshock were about 50 %of the computed widths. The post-reshock thicknesses agree well with the experiment,table 4, with discrepancy of the order of 4 % for the Mach 1.50 and 1.98 casesand 10% for the Mach 1.24 case. Note that t= 0 in the experiments corresponds


14/33


++

++++++++++++++++++++++++++++++++

+

++

+

+

+

+

+

+

+

+

+

+

+

+

++++++++++++

0.16

0.14

0.12

Case VIIb

Case VIb

Case IIb

0.10

0.08

0.06

0.04

0.02

0 5 10

t(ms)

Mixingzonewidth(m)

15

Figure 4. The evolution of the mixing-layer width MZ. The growth rates inferred from theexperiments are indicated as straight lines with the appropriate slopes drawn in the approximatetimes over which they were observed. See table 4 for Mach numbers.

to the first shock interaction, but in simulations, the shock is initialized at 5 cm tothe left of the interface; hence, the offset in measurement times. The much better

agreement after reshock is consistent with the detailed Mach 1.50 experiment of Vetter& Sturtevant (1995) (summarized in their figure 8) in which the initial conditionswere varied by altering the membrane and wire mesh configuration. In the case oftwo different initial amplitudes, they found very similar post-reshock behaviour andnearly identical widths were recorded.

Figure 5 shows the computed mixing width for the higher resolution Mach 1.50run where the experimentally measured growth rates of Vetter & Sturtevant (1995)are also indicated. Both shockinterface interactions are apparent in the form ofsharp compressions of the mixing zone. These compressions are followed by periodsof mixing-zone growth. The interaction of an expansion with the growing mixinglayer can also be observed in the slight change of the layer growth rate at t

4.7ms.

This expansion is generated by the reshock event, after it reflects from the wall, andimpinges on the mixing layer (see figure 2). The evolution of the mixing width afterreshock can then be divided into distinct parts: the initial growth from the reshockwhich starts to decay around 4.5 ms, further growth stimulated by the reflected-expansion event, and peaking at 6 ms followed finally by turbulent saturation and asubsequent very slow period of growth. These stages in the life of the post-reshockturbulent mixing zone can be seen clearly in the turbulent kinetic energy of the flow.

5.2. Turbulence statistics

Planar-averaged statistics of the different flow fields for the case with initial shock

Mach number of 1.50 are reported here. Results are presented at two times: t= 1 m sshortly after the initial shock interaction while the mixing layer is initially growingbut not fully turbulent, and t= 10 ms after the reshock and expansion wave events ata time when the mixing zone is fully turbulent. The Favre-plane-averaged turbulence


15/33


0 2 4 6 8 10

0.02

0.04

0.06

0.08 37.2 m s1

4.2 m s1

0.10

0.12

Mixingzonewidth(m)

t(ms)

Figure 5. The evolution of the mixing-layer width MZ. The experimentally measured valuesare indicated as are the approximate times in which they were observed.

statistics of interest include the resolved-scale turbulent kinetic energy

K

=

1

2uk uk

ukuk2 , (5.4)

the resolved-scale turbulent dissipation

res = dij S

ij

, (5.5)

the subgrid turbulent kinetic energy

k = kk2 , (5.6)

and the subgrid energy transfer

sgs =

ij Sij

, (5.7)

where primes denote fluctuations with respect to the plane-average. The total turbulentkinetic energy is estimated then as K = K + k and the turbulent dissipation as = sgs + re s. These quantities are used to define the turbulent intensity,

u =

2K

3, (5.8)

and the turbulent Mach number

Mt =u

c , (5.9)


16/33


10 (a)

8

6

Density(

kgm3)

4

2

0 0.2x

0.4 0.60

0.4

0.8

1.2

1.6

2.0

10 (b)

8

6

Ratioofspecificheats,

4

2

0 0.2x

0.4 0.60

0.4

0.8

1.2

1.6

2.0

Figure 6. The (y, z)-plane averaged density (solid line) and ratio of specific heats (dotted line) profiles (a) t= 1 ms, during the evolution of the flow. (b) t=10ms.

using the average of the speed of sound c =p/. A turbulence integral scale canalso be defined as

=u3

, (5.10)

allowing the calculation of the turbulent Reynolds number as

ReT =u . (5.11)

In figure 6, the density at time t= 1 ms shows a small reflected shock near x =0.2 mtravelling to the left. The contact, which will evolve into the mixing zone, can be seenin both the density and the ratio of specific heats near x = 0.15 m following behindthe transmitted shock (x = 0.225 m) as both travel to the right. In the later timeplot, t= 10 ms, the shock has reflected off the closed end of the tube at x = 0.62m,reshocked the mixing zone and left the domain. Here, the much larger extent of themixing zone is clearly visible. The shocks and expansion interactions with the mixingzone are inherently compressible effects and form the principal mechanisms for thedeposition of vorticity in the mixing zone.

One useful feature of the stretched vortex subgrid model is its ability toestimate directly local subgrid quantities such as the local subgrid kinetic energyk and the energy transfer off-grid. Figures 7 and 8 show that

k

is of the order of 10

to 20 % of the resolved counterpart and, as expected from a proper LES, the planar-averaged subgrid energy transfer, sgs is about 10 times the resolved dissipation priorto reshock and 100 times larger after reshock. This is another indication that theturbulence scales, down to the unresolvable range, fully develop only after reshock.

Figure 9 shows profiles of turbulent Mach number at two different times. It isobserved that the relative effects of compressibility during the evolution of the mixingzone peak at its centre. While the plot at t= 1 ms has a much lower overall turbulentMach number, the larger peak of Mt = 0.07 in the post-reshock, t=10ms, stage isclearly in only the weakly compressible range. Thus, excluding the shock interactionevents, compressibility effects in the turbulence are not large for this simulation.

The length scale and turbulent Reynolds number ReT are presented in a timerange 7 ms 10 ms which starts shortly before the energy cascade forms, as indicatedby the development of the k5/3 scaling by time t= 7.6 ms, shown next. In figure 10,it can be seen that the length scale and Reynolds number decay during this periodat the centre of the mixing zone as the large bubbles created by reshock interact and


17/33


Subgri

dKE

0

1

2

3

4

5

6

x

Resolve

dKE

00 0.2 0.4 0.6

x0

0 00.2 0.4 0.6

10

20

30

40

50

60

(a) (b)

Subgri

dKE

10

20

30

40

50

60

ResolvedKE

100

200

300

400

500

600

Figure 7. The (y, z)-plane averaged resolved scale turbulent kinetic energy (solid line) com-pared with the subgrid kinetic energy as estimated by the model (dotted line). Note the orderof magnitude difference in the scale of the plots. All quantities in MKS units. (a) t=1ms,

(b) t=10ms.

Resolveddissipation

00 0.2

x x0.4 0.6 0 0.2 0.4 0.6

50

100

150

200

250

300(a) (b)

Subgriddissipation

0

500

1000

1500

2000

2500

3000

Subgriddissipation

0

5000

10000

15000

20000

25000

30000

Resolveddissipation

0

50

100

150

200

250

300

Figure 8. The (y, z)-plane averaged resolved scale dissipation rate (solid line) compared withthe subgrid dissipation as estimated by the model (dotted line). All quantities in MKS units.(a) t= 1 ms, (b) t=10ms.

x

TurbulentMachnum

ber,Mt

00 0.2 0.4 0.6

x0 0.2 0.4 0.6

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

(a) (b)

Speedofsound(m

s1)

0

100

200

300

400

500

Speedofsound

(ms

1)

100

200

300

400

500

TurbulentMachnu

mber,Mt

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Figure 9. The (y, z)-plane averaged resolved scale turbulent Mach number (solid line)

compared with plane averaged local speed of sound (dotted line). (a) t= 1 ms, (b) t=10ms.

compete. It is noted that is comparable to the 2 cm spacing in the initial interfaceperturbation and although K has dropped substantially by t=7ms, ReT is still inthe range of 30 000 to 100 000.


18/33


Time (ms) Time (ms)

Integrallen

gth(m)

7 8 9 10 7 8 9 10

0.05

(a) (b)

0.04

0.03

0.02

0.01

Reynoldsnumber

20 000

40 000

60 000

80 000

10 0000

Figure 10. (a) The plane-averaged integral length scale, l, and (b) turbulent Reynolds

number, ReT, computed in the centre of the mixing zone as a function of time.

Time (ms)

|K|

+|k|

2 4 6 8 100

20

40

60

80

100

~

~

Figure 11. Volume-averaged total turbulent kinetic energy |K|+ |k| as a function of time forthe M= 1.50 case.

Finally, a global turbulence measure can be obtained from the volume-averagedturbulent kinetic energy. The total turbulent kinetic energy deposited by the shock-contact interactions as well as expansion fan-contact interactions can be measuredusing this statistical quantity. Figure 11 shows the total amount of energy deposited bythe initial shock, visible as a very small bump close to the time origin, as well as thatowing to the reshocking event, at 3.5 ms. Following a steep decay in energy formingthe first stage in the post-reshock mixing zones life, a subsequent interaction with

the expansion fan, shown in the wave diagram (figure 2), deposits a relatively largeamount of energy over the duration of approximately 1 ms, peaking near 6 ms. Thislast vorticity deposition corresponds to the second period of post-reshock growth.After these events, there is a very slow period of decay since none of the additional


19/33


k5/3

k

Eu2

D(

k)

103

102

101

100

100 101 102

10

1

102

103

Figure 12. Radial power spectra of velocity E2Du (k) computed in the centre plane of the TMZat four different times: t= 4.5 ms (dotted line), t= 6.5 ms (dashed-dot line), t= 7.6 ms (dashedline) and t= 10 ms (solid line). All computed wavenumbers shown and kmax = 128.

weak expansion fans and compression waves coming from the wall posses a sufficientlylarge pressure gradient to deposit noticeable amounts of vorticity. It is notable thatthe first expansion fan deposits substantial amounts of vorticity, and kinetic energy.This energy is comparable to that of the re-shocking event and occurs becausethe expansionfan interaction takes place over a longer time period during which themixing zone is rather thick, with a wide range of spatial scales and gradients in thedensity field.

5.3. Velocity, density and scalar spectra

The simulation domain was designed with periodic boundary conditions in the cross-section of the tube, in part to allow for two isotropic directions in the flow andto enable the calculation of instantaneous radial spectra. The radial spectra of anarbitrary function f(y, z) is defined as

E2Df (k) =12

k 2

0 |F(k, k)

|2 dk , (5.12)

where F is the Fourier transform of f in polar wavenumber space with k andk denoting the radial and azimuthal wavenumbers. In practice, this is most easilycalculated in Cartesian wavenumber space ky , kz by annular bin-counting. The spectraof the individual velocity components u = u1, v = u2, w = u3 as well as of the density and scalar was calculated using (5.12).

The radial velocity spectra was always calculated at the plane located in the centreof the mixing zone. The results shown in figure 12 indicate that the spectra assumesa persistent k5/3 scaling after the passage of the expansion fan and a reorganizationof the deposited kinetic energy. The expansion interaction starts at about 4.7ms and

continues until 6 ms, as seen in the wave diagram figure 2 and in the plot of thetotal turbulent kinetic energy figure 11. The Kolmogorov-like k5/3 energy spectradevelops by t= 7.6 ms and persists for the remainder of the simulation. Similarly,the density and scalar spectra also develop a persistent k5/3 scaling by this time.


20/33


k

k5/3

E2

D(k),E2

D(k)

106

105

104

103

100 101 102

102

101

100

Figure 13. Radial power spectra of density (solid line) and mixture fraction (broken line),i.e. E2D (k) and E

2D (k), at t= 10 ms computed in the centre plane of the TMZ. All computed

wavenumbers shown: kmax = 128.

x

Densityvariance

Scalarvariance

V

elocityvariance

00 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6

1

2

3(a) (b) (c)

0

50

100

150

200

0

0.01

0.02

0.03

0.04

x x

Figure 14. The x-plane variance of the resolved scale density, mixture fraction, and velocityat t=10ms. (a) 2 2, (b) u2i / ui2/2, (c) 2/ 2/2.

Figure 13 demonstrates that while scalar and density are not uniquely related ( isconstrained mathematically to be 06 6 1 and obeys a different governing equation)their spectra correlate well. The highest resolved wavenumbers in figure 13 showminor effect of aliasing errors. We emphasize that no explicit filtering of any kindwas performed in the present LES and WENO is not used in this region of the flow.

5.4. Mixing statistics

The mixture fraction field parameterizes the degree of mixing of the two gases.Although the flow at late times is quite turbulent, mixing statistics display inhomo-geneities associated with the non-isotropic direction x at all times during oursimulations. Physically, this is related to the presence of different gasses and thevery different bubble-spike structures characteristically observed on either side of themixing zone. We examine the statistics of the mixture fraction in planar cuts takenfrom the centre of the mixing zone, xc , as well as from planes one quarter of the

mixing width MZ from the centre on either side xq = xc MZ/4.The variance for the density, velocity and mixture fraction, figure 14, highlight

the action of the large scales in the flow. Although the density variance is clearlystronger on the SF6 side, the highest turbulent velocities are found closer to the centre


21/33


of the mixing zone. The two peaks in the mixture fraction variance correspond todominantly bubble and dominantly spike portions of the mixing zone separated by aregion of low variance. The energetic structures penetrating into air from the mixingzone produce much larger mixture fraction variances.

To provide a more complete picture of the mixing evolution of the two gases,

we investigate now the behaviour of the mixture fraction p.d.f. For variable densityflows, it is natural to employ the Favre p.d.f. (Bilger 1977), formally obtained fromthe Reynolds joint density-mixture fraction p.d.f., P(, ; x, t), through,

P(; x, t) =1

P(, ; x, t) d, (5.13)

where the independent variables and in the p.d.f.s denote the sampling variablesin this context, not the LES fields. Moreover, we remark that, strictly speaking, P andP represent the p.d.f.s of resolved field quantities when obtained from the LES. Theydo not denote the p.d.f.s of the total fields, which would include the information of

the unresolved scales in the LES. We defer the question of the effect of the unresolvedscales on the p.d.f. to 7.3.

The Favre p.d.f.s, P(), as a function of time are shown in figure 15 for thethree planes in the mixing zone previously specified. These p.d.f.s were formed byconstructing the histograms of resolved mixture fraction, , from the LES at planesof constant x. These results have been obtained at times that correspond, roughly,to instants just before reshock, just after reshock, just past the peak of kineticenergy deposition by the rarefaction-wave interaction and the end of the simulation,respectively. In the initial phase before reshock (figure 15a), little mixing has takenplace. This is evident from the fact that the planar P() are very intermittent and mostof the fluid is composed of unmixed gases. The two large peaks at the extreme valuesof mixture fraction, 0 and 1, are evidence of this state of the gas. This is the phasewhere the inviscid linear and nonlinear instability mechanisms of RMI dominate;physical and subgrid diffusion have not had sufficient time to act. Almost immediatelyafter reshock (figure 15b), very fast mixing caused by the vorticity deposited in themixing zone by the shock leads to a P() with a strong central mode for the planeat xc. The structure of P() at the other two planes close to the pure gases havealso changed. The two intermittent peaks at the extreme values of mixture fractionhave moved towards the centre of the figure and are now less pronounced. Someadjacent broadening of P() including the appearance of plateau is also observed.Before the interaction with the rarefaction wave becomes visible (figure 15c), P() at

the plane in the centre and that close to the SF6 side become even more unimodal,with peaks that are well correlated with the average value of mixture fraction at thatplane. The p.d.f. at the remaining plane, air side, still preserves a strong degree ofintermittency with large amounts of unmixed air. Long after the interaction with therarefaction wave and the subsequent development of the energy cascade (figure 15d),the most salient features of P() have not changed. We do observe that the width,or variance, of P() has narrowed somewhat; indicating further mixing progress. Asalient feature of the curves for the central plane and that close to the SF6 side is thatthe enhanced mixing is non-uniform, or rather forced. We observe the developmentof two small peaks diametrically opposed to the location where we had seen a single

peak. Our observation is that the rarefaction wave deposits more energy on thosephysical regions of the domain where the density gradient is largest. While P() itselfis a single-point quantity and thus deprived of scale information, it is reasonableto speculate that the largest density gradients will exist in the region between the


22/33


0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

1

2

3

4

5 (a)

(c)

(b)

(d)

0

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

Figure 15. Probability density function P() of the mixture fraction at different timescalculated from the resolved scales only, across three planes of the mixing zone: xc 0.25MZ(dotted line), xc (continuous line) and xc + 0.25MZ (dashed line). (a) t=3ms, (b) 3.6ms,(c) 7 ms, (d) 10ms.

unmixed gasses and the relatively well-mixed core of the TMZ. Then, more vigorousmixing should take place in what appears to be the tails of P(). This bimodalcharacter of the central P() persists as the bubbles and spikes continue to transport

partially mixed gases from the outer regions of the mixing zone. We do expect thatas time advances further, the bimodal character of P() should evolve into a singlemode since the remaining rarefaction waves emanating from the wall are very weak.

6. Subgrid continuation

The structural nature of the stretched-vortex subgrid model can be used to developa multiscale treatment of the subgrid activity. This is done in a way which is fully con-sistent with its implementation when determining the subgrid fluxes in LES. Velocityspectra, including the anisotropic components, can be estimated directly for scales

below the cutoff scale from the formulae previously presented. Subgrid continuation ofthe scalar spectra requires one additional assumption to estimate the effect of Schmidtnumber, and subgrid mixing statistics require further assumptions of the form of thesubgrid p.d.f. Here we outline the mechanics for calculating these continuations in the


23/33


context of the velocity spectra. In the subsequent sections, the additional modellingassumptions required for the scalar are discussed and we present results.

The two-dimensional velocity spectra may be calculated in terms of kr , the radialcomponent of wavenumber vector k= (k1, k2, k3) in polar coordinates. This is a lengthycalculation which closely follows the detailed derivation of the one-dimensional

spectra given by Pullin & Saffman (1994). For that reason, only the main results areoutlined here. We begin with the two-point velocity and vorticity correlation tensors

Rij (r) = ui (x)uj (x + r), (6.1a)Wij (r) = i (x)j (x + r), (6.1b)

where denote ensemble averaging and we omit the tilde notation of the velocityvector since we are dealing with the complete velocity vector. The correspondingFourier transforms are given by

Rij (r) = ij (k) exp(ik r) dk1 dk2 dk3, (6.2a)Wij (r) =

ij (k) exp(ik r) dk1 dk2 dk3. (6.2b)

Pullin & Saffman (1994) have shown that without assumptions of isotropy, thevorticity correlation tensor may also be expressed as

Wij (r) =1

2V

m

lm

0

20

20

|m(1, 2, t)|2

exp(

i1r

1 i2r

2)U3i U3j

P(

,

,

) d1 d2 sin

d

d

d

, (6.3)

where the sum is over an ensemble of vortices, length lm, whose orientations withrespect to the laboratory frame of volume V are described by the p.d.f., P, and theEuler angles , , . The matrix Uij is a unitary rotation operator that maps vortex,r j , and laboratory coordinates, and m(1, 2) is the Fourier transform of the vorticityexpressed in the frame of the vortex.

An expression for the two-dimensional energy spectrum may be arrived at by firstdefining the two-dimensional energy tensor as

E2Dij (kr ) =kr

2

0 2

0

ij d dk3, (6.4)

where k1 = kr cos , k2 = kr sin and k2r + k

23 = |k|2. From the relationship between the

transform of the velocity and vorticity correlations ij and ij , we obtain

ij (k) = qq (k)(|k|2ij ki kj ) |k|2j i , (6.5)which allows the two-dimensional velocity spectrum to be expressed in terms of thevorticity correlation transform

E2Dij (kr ) =kr

2

0 2

0

1

|k

|2

qq (k)ij ki kj

|k

|2

j i (k)d dk3. (6.6)

Using the inverse Fourier transform of (6.2) and the alternative expression for Wij ,(6.3), in (6.6), the two-dimensional energy tensor E2Dij leads, by integration of resultingdelta functions, to


24/33


E2Dij (kr )

=kr

V

m

lm

20

0

|kr /cos |kr

20

||2

(ij ki kj 2 U3i U3j )P(, )

2 k2r

1/2

k2r 2 cos2

1/2

d d d,

(6.7)

where 2 is found to be equivalent to k2, and P has been assumed to be independentof the spin angle . The two-dimensionally shell-summed energy spectrum Eii cannow be related to the three-dimensionally shell-summed energy spectrum by using thesum identity U3i U3i = 1 and the result (Lundgren 1982) that the shell-summed energyspectrum is related to the transform of the vorticity for an ensemble of vortices by

E() =22

V

m

lm

20

1

|m(1, 2, t)|2 d, (6.8)

to arrive at

E2Dqq (kr ) =kr

22

20

0

|kr / cos |kr

E()P(, )sin 2 k2r

1/2k2r 2 cos2

1/2 d d d. (6.9)For the simple vortex alignment model used in the stretched vortex model, the p.d.f.has the form

P(, ) = 4sin o

( o)( o), (6.10)which results in

E2Dqq

(kr) =

2kr

|kr / cos o |

kr

E() 2 k2r 1/2k2r 2 cos2 o1/2 d. (6.11)The component of the shell-summed energy spectra that results from the out-of-planedirection E2D33 may be calculated directly by using the relation k

23 = k

2 k2r and thefact that U33 = cos

, giving

E2D33 (kr ) =2kr

|kr / cos o|kr

k2r 2 cos2 o

1/2E()

2

2 k2r1/2 d. (6.12)

At an instant in time, the velocity spectra for the resolved fields may be calculated asradial autocorrelations in a given plane of constant x, exploiting the periodic boundaryconditions of the flow. The subgrid continuation of the velocity spectra is obtainedby averaging the two-dimensional subgrid spectra over all the cells of the planeand the two-dimensional velocity spectra is in turn related to the known three-dimensional spectra (3.12) for a given vortex by (6.11). The spectra of the velocitycomponent normal to the plane (the u-velocity) is given by (6.12). In turn, this impliesthat the remaining two velocity components v, w, whose directions lie in the plane,contribute E2Dv +E

2Dw = E

2DE2Du . Such spectral continuations allow the presentationof anisotropic and isotropic portions of the total (resolved + subgrid) velocity spectra,as will be shown in 7.1.

7. Subgrid continuation statistics

We use the subgrid continuation ideas to develop consistent estimates of subgridspectra for the velocity and scalar fields. In this context, the results show that the


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k k

E(K)/(v5)1/4

2/3k5/3E(K)

108

(a) (b)106

104

102

100

102

104

106

101

102

103

104

105

106

107

108

106 105 104 103 102 101 100 101 0.2 0.4 0.6 0.8 1.0

(k)5/3

Figure 16. The radial spectra (solid line) with computed subgrid continuation (broken

line) at time t= 10 ms computed in the centreplane of the TMZ. (a) Normalized spectra;(b) compensated spectra.

resolved spectra can be extended consistently into the subgrid. In the case of a mixingscalar, the continued spectra can incorporate any effects owing to non-unity Schmidtnumber, Sc = /D. Then, appropriate integration of the spectra generates estimatesof subgrid scalar moments. In turn, these scalar moments can be used to constructassumed subgrid p.d.f.s of the mixing field of any reasonable complexity. For example,the resolved scalar field and second-order moment (variance) can be used to constructsimple subgrid presumed p.d.f.s (Gaussian, beta, etc.). Presumed p.d.f.s with three or

more moments can also be constructed with some additional modelling (Effelsberg &Peters 1983; Mellado, Sarkar & Pantano 2003).

7.1. Turbulence spectra

In the general practice of LES, it is not possible to obtain estimates of the fullturbulence energy spectra, and consequently there is limited information to judgewhether the resolved cutoff scale of the simulation reaches well within the inertialsubrange (a theoretical premise of LES). Estimates of the complete turbulence spectraalso enable more accurate calculation of the turbulent dissipation and hence, of theKolmogorov length scale, = (3/)1/4. With this in mind, our subgrid model andresolved geometry were chosen to allow such estimates to be made. Since the flow

possesses two homogeneous directions, an averaged for each plane of constantx can be obtained by using the plane-averaged values of . Figure 16(a) showsthe normalized and figure 16(b) the compensated resolved and continued turbulencespectra in the plane of constant x at the middle of the mixing zone for t=10ms. Asexpected, the roll-off for the spectra begins near k = 0.1, consistent with the majorityof the dissipation occurring in the scales k > 0.1. Note that this estimate is not trivial.From (3.12), it is seen that the viscous decay of the spectrum of the stretched-spiralvortex is Gaussian, a natural consequence of the second-order viscous operator in theNavierStokes equations. In contrast, figure 16(b) indicates that the present estimateof the plane-averaged subgrid energy spectrum decays exponentially when k = O(1),

in agreement with experimental evidence, Saddoughi & Veeravalli (1994). The changefrom local Gaussian (in a cell) to plane-averaged exponential decay is explained bythe influence of the statistics of fluctuating axial strain along the subgrid vortex axis(the quantity |a| in (3.12)); for analysis, see Pullin & Saffman (1993).


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K

101

(a) (b)100

100 101 102 103 104 105 100 101 102 103 104 105

101

102

103

10

4

105

106

107

k5/3

[Ev2

D(k)+Ew2

D(k)],Eu2

D(k)

Eu2

D(k)/E2D(k)1/3

K

0.4

0.2

0.2

0.4

0

1 2

Figure 17. Components of the radial spectra at time t= 10 ms computed in the centreplane ofthe TMZ displaying resolved scale (solid line) and continuation (broken line). In (a) the spectrais divided into the anisotropic E2Du (k) (upper curve) and isotropic (1/2)[E

2Dv (k)+E

2Dw (k)] (lower

curve) directions. In (b) a measure of the anisotropy is displayed as E2Du (k)/E2D(k) 1/3.

As discussed in the previous section, it is now possible to use the stretched-vortexmodel to derive estimates of the isotropic and anisotropic parts of the spectra withthe present multiscale approach. Figure 17(a) shows the anisotropic, u component ofvelocity E2Du , and the isotropic radial spectra of velocity components, E

2Dv + E

2Dw . In

this figure, we show the spectra obtained from the resolved velocity fields (continuousline) as well as the subgrid continuation part (broken lines) at the same location usedin figure 16. The continued spectra was obtained by computing the two-dimensionalradial spectral, (6.11)(6.12), for each cell in a plane and then averaging the resultfor the entire plane. For comparison, we also show the 5/3 constant slope linein the same figure. It can be noted that this LES is conducted at a resolution wellwithin the inertial subrange, since we recover more than a decade of wavenumberswith a Kolmogorov-like spectra in the resolved field. Figure 17(b) gives a scale-dependent measure of the anisotropy that the model continues into the subgrid. Thisindicates that E2Du contains, at almost every scale, more than a third of the totalenergy. The spectra detailed from the simulation is reported unprocessed, that is, allwavenumber contributions are shown and no filtering is performed at the highestwavenumbers. It is our experience that the use of skew-symmetric-type discretizationtogether with bandwidth-optimized centred stencils, like the present TCD, leads tovery good spectral results with minimal accumulation of energy owing to aliasingerrors at the high-wavenumber end of the spectrum.

7.2. Effect of Schmidt number on continued scalar spectra

We remark from the outset that we will allow the ratio of molecular diffusivity toscalar diffusivity, known as the Schmidt number, to take a series of values that aredifferent from the actual ratio corresponding to our gas mixture. This is done in thespirit of exemplifying the methodology for other gases with different diffusivities andis carried out in both this subsection and the following one.

Pullin & Lundgren (2001) presented an approximate solution to the equationsdescribing the mixing of a passive scalar inside a stretched-spiral vortex. They obtainedthe scalar spectrum (their equations (99)(106)), which, for the scalar , we express

in the form

E (k) = K

k5/3exp

(4 + 2D)k

2

3a

+

8

5

2

a

1/3k1exp

2Dk

2

3a

, (7.1)


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k

E2

D(k)

102

104

106

108

1010

1012

1014

100 102 104 106 108 1010

k1k5/3

Figure 18. The time t= 10 ms radial spectra of the scalar computed in the centre plane of

the TMZ with subgrid continuation (broken lines) assuming, from left to right, Sc =1, 103

,106, and 109.

where a is the axial resolved-scale strain rate along the subgrid vortex axis, isthe subgrid vortex circulation and K is a group prefactor whose numerical valuedepends on physical quantities that describe the internal structure of the vortex andthe initial scalar field. Equation (7.1) is consistent with phenomenological models forscalar mixing in turbulence (see Tennekes & Lumley 1974). When the scalar diffusivityD , there is a rapid decay like k5/3 through the inertial convective range, withslower decay as k1 before reaching diffusive cutoff at the Batchelor scale, while fordiffusivities D = O() the cutoff occurs before the k1 range begins. The derivation ofthis form is consistent with the subgrid scalar flux (3.3d); the mixture fraction withina computational cell is imagined to be wound by an elemental Lundgren subgridvortex. The group prefactor K is computed here for each cell by dynamic structurefunction matching, similar to that outlined in 3.2 for the subgrid velocity spectrum.This spherically averaged structure function formed by evaluated from the resolvedscale at the length scale = x, F2 (), is equated with the subgrid expression ofthe same quantity

F2 () = 4K

2/3

0 s5/3 +

8

52

a2

1/3

s11sin s

s ds. (7.2)For our model, the remaining parameter is the subgrid circulation /, which, from

(7.1), largely determines the relative weighting between the k5/3 and the k1 spectralcomponents. Pullin & Lundgren (2001) found that / = 1000 gave fair agreementwith experimental data at Sc > 1, and we adopt this value here. This is the onlyadditional assumption required for the subgrid continuation of the scalar spectrum.It may seem paradoxical that Sc dependence at small scales can be inferred from LESin which Sc does not appear explicitly. This is because we implicitly assume, in theLES, that the resolved scale cutoff c lies somewhere in the inertial range, and that,consequently, the flux of scalar variance off the resolved-scale grid is independent

of Sc for Sc > 1. This is consistent with the hypothesis that mixing is an essentiallysmall-scale process that is subgrid in the present LES.

Figure 18 shows continuation of the resolved scale composition spectra into thesubgrid for different Schmidt numbers. As in 7.1, this was done by computing


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the two-dimensional radial spectral continuation for each cell in a plane and thenaveraging the result for the entire plane. As can be seen, the large Sc spectra displayboth the k5/3 and k1 range. While the tails of the spectra can be quite long for thishigher Schmidt number, its contribution to the subgrid scalar variance is typicallynot large, unless the Schmidt number is huge.

The subgrid variance of the composition scalar 2 within a given cell may becomputed from the model

2 = 2

kc

E (k) dk, (7.3)

where kc is the cutoff wavelength associated with the grid spacing and the results fromthe structure function matching, which implicitly assumes 2

0

E dk =V1 | |2 dx.By examining the relevant portion of (7.1), it is easy to see that there is only logarithmicdependence on Schmidt number for the subgrid contribution to the total variance,that is

kc

s1 exp2Ds 23a

ds = 12 Dk 2c 2/(3a)

s1 exp(s) ds, (7.4a)

= log( )

n=1

(1)n n

n(n!), (7.4b)

where = Dk2c 2/(3a) and = lim0+ (ln 1

s1 exp(s) ds) 0.5772 is Eulers

constant. For fixed and large Sc then, it can be seen that the leading behaviouris log(Sc) log(2k 2c /(3a)) . This result is also consistent with the argumentspresented by Dimotakis (1989) that account for the Schmidt-number dependence ofthe total scalar variance.

7.3. Effect of Schmidt number on continued p.d.f.

Actual p.d.f.s (regular or subgrid) are not generally known in LES. They containthe complete single-point statistics of the state of the mixture. Two approaches arecommonly used in LES to estimate or model the subgrid p.d.f.: solve a subgridtransport p.d.f. equation (Colucci et al. 1998) or directly presume a functional formwith specified moments (Cook & Riley 1994). Here, we take this latter approachfor simplicity, but the alternative of solving a transported subgrid p.d.f. equation isalso feasible. Using results from previous sections, it is now possible to constructa presumed subgrid p.d.f. that can take into account possible non-unity Schmidt-

number effects. We consider the subgrid Favre p.d.f. of mixture fraction, Psgs(), toclarify ideas.

There are typically two uses of Psgs. In the first case, the physics of the problemare such that there is two-way coupling between the scalar and the flow and we mustobtain subgrid expectations of some nonlinear function of for LES closure; let uscall this function f(). Then, knowledge of Psgs gives, by direct integration,

f(; x, t) = f()Psgs(; x, t) d. (7.5)This is typically the closure of LES for combustion. In the second case, we may be

interested in mixing per se. The question of interest now is whether we can estimate thetotal scalar p.d.f., P(), (resolved and subgrid) from Psgs . To expose the relationshipbetween these two p.d.f.s, we must recall Gao & OBrien (1993) and introduce thefine-grained p.d.f., ((x, t)), where (x) is Diracs delta function, is the sample


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space variable and (x, t) is a realization of the random spacetime varying field.We show the derivation of this relationship for the Reynolds and Favre p.d.f.s. Theregular and subgrid p.d.f.s are obtained from the fine-grained p.d.f. through

P(; x, t) =

(

(x, t))

, (7.6)

Psgs(; x, t) = G(x x)( (x, t)) dx, (7.7)where denotes ensemble averaging and G(x) is the convolution kernel as in(3.2). Now, ensemble averaging (7.7), commuting the spatial and ensemble-averagingoperators (both are linear operators), leads to

Psgs =

G(x x)( (x , t)) dx

=

G(x x)P(; x , t) dx . (7.8)

We further observe that: by construction, G only affects the scales in the

neighbourhood and below the subgrid cutoff scale c and that the spatial variationsof P happen, in general, on scales of the order of the integral scale . Then, wecan take the p.d.f. outside the integral (using the normalization property of the filterkernel) giving

G(x x)P(; x , t) dx P(; x, t). (7.9)The approximation in (7.9) is quite accurate because the error, that is at most oforder c/, is small in LES.

The equivalent of (7.8) for the Favre version of the p.d.f. is

Psgs = G(x x )(x, t)( (x, t)) dx=

G(x x )

( (x, t))( (x, t)) d

dx

=

G(x x )

P(, ; x , t)d

dx

=

G(x x )(x, t)P(; x, t) dx . (7.10)

In the last term, we have removed the superscript from the ensemble average of ,since there is no longer any ambiguity with the sample variable. Using the approxima-tion introduced in (7.9) and neglecting the additional variable density contributionsresulting from filtering the product of the ensemble-averaged density with the Favrep.d.f., we arrive at

G(x x)(x, t)P(; x , t) dx (x, t)P(; x, t). (7.11)

To clarify ideas, we now introduce a concrete function Psgs. Among the possiblechoices of presumed p.d.f.s, the beta distribution is very popular (Cook & Riley 1994).

For this model, Psgs() within a given cell is assumed to take the formPsgs(; x, t) = (a + b)

(a)(b) a1(1 )b1, (7.12)


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0.2 0.4 0.6 0.80

1

2

Figure 19. Effect of Schmidt number on continued p.d.f. of mixture fraction calculated in thecentre of the mixing zone at t= 10 ms with resolved scale p.d.f (solid line) and values Sc = 1,107 (dash-dot and broken line, respectively).

where (x) is the gamma function. The parameters a and b are determined byevaluating the first- and second-order moments of Psgs, leading to the followingrelationships with the resolved scales: a = [(1 )/2 1] and b = a(1/ 1).Recall, the LES gives directly, and the scalar variance can be estimated from(7.3). For clarity, we avoided denoting the dependence of a and b on space and timeexplicitly, but note that

Psgs varies with position and time parametrically through the

dependence of a and b on the resolved scalar and scalar variance fields.The formal method for computing the estimate for P in the mixing zone consist

in noticing that (7.12) is in reality the subgrid conditional-p.d.f. Psgs ( |, 2 ). UsingBayes theorem, the subgrid joint-p.d.f. is now given by

Psgs

, , 2 ; x, t

= Psgs

|, 2Psgs, 2 ; x, t, (7.13)

and the subgrid p.d.f. of is given by

Psgs(; x, t) =

Psgs

|, 2Psgs

, 2 ; x, t

d d2 . (7.14)

Then, P results from the integration and ensemble averaging given by (7.11), that forour flow is expressed as the planar-averaged expression

P(; x, t) Psgs (; x, t)(x, t) , (7.15)

where (x, t) has been approximated by (x, t). In practice, these steps are perfor-med simultaneously by computing the average of the conditional-p.d.f., Psgs( |, 2 ),across the cells of the plane in question, as shown by Jimenez et al. (1997) for theconstant density case.

Figure 19 shows the effect of a range of Schmidt numbers on the estimate of thetotal p.d.f. taken in the centre of the mixing zone at one time. It can be seen thatas the Schmidt number increases, the logarithmic dependence of the variance on theSchmidt number results in relatively small, but still visible, changes in the overall


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p.d.f. This is because the subgrid cutoff scale is well within the inertial subrange inour simulation so that the contribution of the subgrid scalar variance is small withrespect to the resolved counterpart. Note also that the effect of Schmidt number isnot monotone. Some regions of the p.d.f. are lifted with increasing Schmidt numberwhile others sink; but, in general, the qualitative shape of the p.d.f. is not altered by

the inclusion of subgrid effects.In 5.4, we postponed the discussion of the effect of the subgrid scales on the

p.d.f. of mixture fraction and contented ourselves with the statistics obtained fromthe resolved scales. At this point, we can now consider the consequences of neglectingthe effect of the unresolved scales within the context of the results of our modelling.Evidently, most of the scalar variance is already represented by the resolved fieldand only a small amount, that could become non-negligible if the Schmidt numberis huge, is present in the subgrid. We can now go back to figure 15 and claim witha degree of certainty that those p.d.f.s are representative of the state of the mixture.Nevertheless, we would like to remark that this statement is based on an estimate

of the effect of the subgrid scales on the p.d.f. using additional modelling, the actualp.d.f. is lost when LES modelling is introduced.

8. Conclusions

The RichtmyerMeshkov instability with reshock was studied in the canonicalgeometry of a rectangular shock-tube with a square cross-section, the gases in thesimulation and the geometry of the domain, and the strength of the incident shockwere chosen to match those of the experiments of Vetter & Sturtevant (1995). Thehigh Reynolds numbers produced within the mixing zone between gases required the

use of the techniques of large-eddy simulation (LES), and the stretched-vortex subgridmodel was employed. A specially constructed hybrid numerical method was used, onethat is conservative and shock capturing, but reverts to a numerically dissipationlessform away from shocks. Of the three different experiments simulated, all had goodagreement with the observed growth rates of the mixing zone after the initial shockand after reshock. The actual width of the mixing zone was not expected to comparewell with the observed values owing to the large degree in uncertainty in the initialconditions, but it was found that the width at the end of the simulation times werebetween 4 % and 10 % of the observed values.

Examination of the turbulent kinetic energy indicated that an expansion wave whichfollows reshock plays a major role, comparab

Documents

D.J. Hill, C. Pantano and D.I. Pullin- Large-eddy simulation and multiscale modelling of a Richtmyer–Meshkov instability with reshock